We divide the lifespan of a tsunami wave into three stages: formation, propagation, and on-shore arrival. For each of these phases, we select a system of equations that captures the appropriate dynamics for that particular phase.

We develop central-upwind schemes for a one-dimensional Savage-Hutter type model of submarine landslides and generated tsunami waves and a non-hydrostatic Saint-Venant system. Our schemes exactly preserve physically relevant steady-states, and preserve positivity of water depth and, where applicable, the schemes are insensitive to choice of discretization of nonconservative products, properly incorporate friction inherent in the model, and properly handle non-hydrostatic pressure terms. We apply our scheme to a variety of test problems and the numerical results clearly demonstrate a high accuracy and robustness of the proposed method. We also demonstrate the importance of the non-hydrostatic pressure terms for wave propagation in the on-shore arrival.